WPS3631
Poverty Traps, Aid, and Growth
Aart Kraay and Claudio Raddatz
The World Bank
World Bank Policy Research Working Paper 3631, June 2005
The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the
exchange of ideas about development issues. An objective of the series is to get the findings out quickly,
even if the presentations are less than fully polished. The papers carry the names of the authors and should
be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely
those of the authors. They do not necessarily represent the view of the World Bank, its Executive Directors,
or the countries they represent. Policy Research Working Papers are available online at
http://econ.worldbank.org.
1818 H Street N.W., Washington, DC 20433, akraay@worldbank.org, craddatz@worldbank.org. This
paper was prepared as a background paper for the 2005 Global Monitoring Report. We would like to thank
Andy Berg, Carlos Leite, and Zia Qureshi for helpful discussions, and Ana Margarida Fernandes for
generously sharing her data on Colombian plants.
Abstract
This paper examines the empirical evidence in support of the poverty trap view of
underdevelopment. We calibrate simple aggregate growth models in which poverty traps can
arise due to either low saving or low technology at low levels of development. We then use these
models to assess the empirical relevance of poverty traps and their consequences for policy. We
find little evidence of the existence of poverty traps based on these two broad mechanisms. When
put to the task of explaining the persistence of low income in African countries, the models
require either unreasonable values for key parameters, or else generate counterfactual predictions
regarding the relations between key variables. These results call into question the view that a
large scaling-up of aid to the poorest countries is a necessary condition for sharp and sustained
increases in growth.
1. Introduction
The idea of poverty traps has captured the imagination of development economists and
policymakers alike for many years. It is not hard to see why. There are many very plausible self-
reinforcing mechanisms through which countries (or individuals) who start out poor are likely to
remain poor. There is also an abundance of empirical evidence that most countries that were
relatively and absolutely poor in the middle of the 20th century remain so today. Recent calls for
across-the-board debt relief and a major scaling up of aid to help poor countries achieve the
Millennium Development Goals have been significantly influenced by the idea that these
countries are stuck in poverty traps and major pushes are required to break free of these traps (see
for example Sachs et. al. (2004, 2005)).
In this paper we examine the empirical evidence in support of the poverty trap view of
underdevelopment. We do this by calibrating specific theoretical macroeconomic models of
poverty traps in order to assess both their empirical relevance and their consequences for policy.
We focus on two specific mechanisms generating poverty traps: low saving and low levels of
productivity at low levels of development. It is not hard to see how either of these can generate a
poverty trap. If either saving or productivity is low at low levels of development, investment will
be low and countries will converge to an equilibrium with low capital and output per capita. If
over some range of income levels saving and/or productivity increase sharply, then if countries
can get to this point they might also be able to converge to an equilibrium with high capital and
output per capita. We focus on these mechanisms because they capture some of the most popular
explanations for the existence of poverty traps, and because they are often used as an explicit
motivation for foreign aid to help countries escape from poverty traps.
We find little evidence of the existence of poverty traps based on these two broad
mechanisms. Admittedly, both mechanisms can theoretically generate poverty traps, or at least
very persistent poverty. However, when put to the task of explaining the persistence of low
income in African countries, the models require either unreasonable values for key parameters, or
else generate counterfactual predictions regarding the relations between key variables. These
results call into question the view that a large scaling-up of aid to the poorest countries will result
in sharp and sustained increases in growth.
In the case of the savings mechanism, generating a poverty trap consistent with the
experience of poor African countries with a simple Solow growth model with a variable saving
rate requires this rate to be a steep s-shaped function of capital per capita. Savings should be low
1
at low levels of capital per worker, increase significantly at some intermediate levels and level off
at high levels. The actual relation between the saving rate and capital per capita observed in the
data is far from meeting these conditions. Instead, saving rates seem to be increasing at low levels
of capital per worker, flat at intermediate levels and increasing again at high levels.
The inability of the savings based explanation to account for the experience of African
countries is not the result of the exogenous behavior of the saving rate imposed by the Solow
model. Building on standard explanations for the inability of countries to save at low levels of
income, we also calibrate a model of subsistence consumption with optimizing agents. We find
that for this model to account for a country's persistent low level of income, the country must be
very close to the level of subsistence consumption. Given the high dispersion of consumption
across countries in Sub-Saharan Africa, accounting for slow growth in these countries requires us
to argue that subsistence consumption levels are country-specific and vary a lot from one country
to the next. If we instead assume a common level of subsistence consumption, the model predicts
that most African countries should be rapidly converging to their stable equilibrium. The model
also predicts that, by rapidly relaxing the constraints that subsistence consumption imposes on
capital accumulation, aid should quickly have an important impact on growth and savings. We
perform a series of very simple regressions and review the existing literature to demonstrate that
this prediction is not borne out by the data.
Technology based explanations of poverty traps based on aggregate models do not fare
better. In a Solow framework similar to the one discussed above for the saving rate, we find that
the relation between TFP and aggregate variables found in the data is significantly different than
the s-shaped pattern required to induce a poverty trap. We also focus on a broad class of models
of optimizing agents facing a technology with increasing returns that are external to the firm.
Within these models, we show that the existence of an stable poverty trap equilibrium requires the
external increasing returns to be sufficiently large relative to the diminishing returns in the
production function. We find that such strong increasing returns are inconsistent with existing
estimates from the vast empirical literature on the estimation of production functions, as well as
some simple estimates of our own.
Based on the previous findings, we conclude that simple aggregate models of poverty
traps we consider are not promising candidates to account for the experience of African countries.
To the extent that these models capture some important features of reality, the results suggest that
a large expansion of aid is not a necessary condition to improve the prospects of these countries.
The available data suggest that, at least at the aggregate level there is not enough non-convexity
2
in underlying fundamentals to expect that large-scale increases in aid will be proportionately
more effective than more moderate amounts of assistance. This does not mean that increasing aid
to Africa (or other poor countries) is a bad idea. Rather, it suggests to us that we should not
expect any disproportionate growth effects of large-scale aid.
We do not claim that this paper provides comprehensive coverage of the possible
mechanisms of poverty traps, and the simple aggregate models we consider here are unable to
capture some possible mechanisms mentioned in the literature. A significant fraction of this
literature is based on highly stylized models, and on non-convexities at the micro-level, which are
not easily amenable to calibration using aggregate data. Also, we face the standard trade-off of
breadth versus depth. We have chosen to analyze a set of broad models that rely on aggregate
mechanisms without delving deeply into the possible micro-mechanisms that lie behind our
aggregate relations. A possible alternative approach would be to take seriously a specific micro-
mechanism of poverty traps and make an effort to calibrate it using micro and macro data.
Nevertheless, we consider that there are some important general lessons that can be derived from
our analysis. The mechanisms that we analyze are usually mentioned in the discussion on poverty
traps, and the models we use are standard in the growth literature. Also, under some conditions
these models can be understood as reduced form representations of the underlying micro-
mechanism.
It is important to notice that by using a representative-agent model in our analysis, we are
implicitly assuming perfect factor markets, which leaves aside an important set of
microeconomic models of poverty traps based on credit and labor market imperfections that result
in apparently large TFP differences across countries.1 However, despite their importance, these
types of models do not speak directly to the issue of whether large across-the-board increases in
aid are likely to move poor countries out of poverty. The reason is that the low aggregate
productivity associated with these types of models is related to distributional problems. A
redistribution of income from rich to poor people within a country could increase measured
aggregate TFP without an increase in aid. Also, modest increases in aid could have large
productivity results if properly targeted, while large increases could have no effect on measured
aggregate TFP if they benefit those that have already moved out of the poverty trap. Therefore, in
1See Banerjee and Duflo (2004) for a survey of these models, and Banerjee (2001) for a detailed
analysis of credit market imperfections.
3
the aid context, these models speak more directly to the importance of aid targeting than to any
non-linear growth effects of aid.
The rest of this paper is organized as follows. In the next section we review some of the
existing evidence on poverty traps. In Section 3 we discuss saving-based poverty traps, and
consider the role of foreign aid in helping countries to escape from such traps. In Section 4 we
consider the evidence for technology-based poverty traps. Section 5 concludes.
2. Existing Empirical Evidence on Poverty Traps
In this section we briefly review some of the few existing empirical studies relating to
poverty traps. One strand of the literature provides very reduced-form evidence based on cross-
country data. If poverty traps are an important feature of growth dynamics, then over time one
would expect to observe a bimodal distribution of per capita income, with a group of poor
countries clustered around the low-level poverty-trap equilibrium, and a group of rich countries
clustered around the high equilibrium (see Azariadis and Stachurski (2004) for details). In a series
of papers Quah (1993a, 1993b, 1996, 1997) has used non-parametric methods to estimate the
evolution of the distribution of per capita income across countries. His results suggest the
emergence of such a bimodal distribution of income. This finding is supported by a recent paper
by Bloom, Canning, and Sevilla (2003) which provides evidence that, after controlling for
countries' geographic characteristics, the evolution of income across countries is consistent with a
bimodal ergodic distribution. There is some controversy, however, regarding the bimodality of
the long-run income distribution: Kremer, Stock, and Onatski (2001) argue that the dynamics of
the world income distribution is better characterized by a prolonged transition, during which
inequality may increase, towards a single-peaked long-run distribution. A similar result is
obtained in Azariadis and Stachurski (2004b), who uses a parametric model of income dynamics
derived from an explicit growth model based on Azariadis and Drazen (1990). They also find that
the long-run income distribution is unimodal, but that bimodality will appear during the
transition. The last two papers are therefore only partly consistent with the idea of poverty traps,
4
in the sense that income disparities may be very persistent and appear as poverty traps, but
eventually, all countries would converge to the same equilibrium.2
On a different front, a recent paper by Feyrer (2003) raises some questions about the
underlying causes behind the finding of bimodal distributions. The paper suggests that the
bimodality of income distribution found by Quah results from a bimodal distribution of TFP, and
therefore from productivity differences. Whether the bimodality of TFP results purely from
increasing returns or from barriers to technology adoption may have different implications about
the validity of the poverty traps story to account for the bimodality of the income distribution.
At a deeper level, the main drawback of this body of evidence is that, for the most part,
their empirical analysis of the evolution of income distribution is non-parametric and unrelated to
any underlying growth model, in particular, to any poverty trap story. The evidence is therefore at
most consistent with a model of poverty traps. It does not provide proof of the validity of these
models, nor does it address whether the magnitudes of the underlying mechanisms are empirically
plausible. An exception is the paper by Azariadis and Stachurski (2004b) cited above, which
assume the presence of increasing returns external to the firm as the underlying cause of non-
convexities. While this model can generate bimodality during the transition, the parameters
estimated for the production function are somewhat unreasonable, suggesting incredibly large
degrees of increasing returns at some levels of income. As we will discuss in section 4, these
levels of increasing returns are not supported by the data.
If poverty traps are important, one would expect also to occasionally see growth spurts as
countries manage to escape from such traps. In a recent paper, Hausmann, Pritchett, and Rodrik
(2004) empirically examine such "growth accelerations", which they define as increases in
growth of at least two percentage points that are sustained for at least eight years. They find that
there are surprisingly many such accelerations, and that while they are in general quite difficult to
predict, it is the case that they are more likely to occur in poorer countries. However, as in the
previous case, this evidence is merely consistent with escapes from poverty traps and it is not
conclusive. As the authors note, the standard dynamics of convergence in a growth model with
shocks would also suggest that large jumps in growth rates are more likely in poor countries than
in rich countries.
2Of course, as pointed out by Quah (2001) the distinction whether poor countries are expected to
remain poor for decades or centuries is nitpicking, and should not deter us from thinking about ways of
accelerating the transition.
5
Overall, this reduced-form cross-country evidence is at most suggestive of the existence
of poverty traps, but it is hardly conclusive. Moreover, this type of evidence tells us little about
the specific mechanisms generating the poverty trap or the policy interventions required to escape
from that poverty trap. More useful in this respect is a line of microeconometric evidence that
focuses on finding direct evidence of the mechanisms underlying specific models of poverty
traps. For example, McKenzie and Woodruff (2004) use data from microenterprises in Mexico to
search for evidence of poverty traps based on non-convexities in the production function. They
take seriously the argument that poverty traps might exist if there are large fixed costs to starting
a business. If capital markets are imperfect and potential entrepreneurs are credit-constrained by
their lack of collateral, then a poverty trap can exist since individuals who start out with low
wealth are unable to finance potentially profitable investments in new businesses. They find that
the fixed costs involved in starting up a small enterprise in Mexico are typically very low, in
some sectors less than half the monthly wage of a low-wage Mexican worker. They also find that
the returns to capital are very high even at very low levels of the capital stock, and cannot reject
the hypothesis that returns are decreasing over the entire range of observed capital stocks. They
conclude that their evidence is not consistent with this particular mechanism of poverty traps.
In contrast, a number of papers have found microeconometric evidence of spillovers or
other externalities that could form the basis of poverty traps. For example, Jalan and Ravallion
(2004) argue for the existence of spatial or geographic poverty traps. In a panel of households
from China, they find that consumption growth at the household level increases with the local
availability of "geographical capital" understood as the availability of roads, the local level of
literacy, etc. Their evidence suggests that these aggregate factors (at the local level) increase the
returns to capital faced by households. However, the main focus of the paper is to determine the
impact of locally aggregate variables on household growth. They do not spell out a feedback
mechanism from household growth to the evolution of the geographical capital that could
generate a poverty trap. A direct interpretation of their evidence is that it shows that the factors
captured in the geographical capital are a constraint to household growth. If we assume that
household growth feeds back into these variables, it might be possible to generate a poverty trap,
but the degree of feedback necessary to induce such a trap is not specified. A different type of
microeconometric evidence can be found in Lokshin and Ravallion (2004), who empirically
estimate non-linear dynamics in household income in two transition economies. While they find
that adjustment to income shocks is nonlinear, they find no evidence of non-convexities that
would cause temporary adverse shocks to permanently lower household income, as would be the
case in a variety of models of poverty traps.
6
Finally, two recent papers take a calibration approach to the study of poverty traps as we
do here. Graham and Temple (2004) use a static two-sector variable-returns-to-scale model and
show how, with minimal data requirements, it is possible to calibrate two steady-states for each
country in the world. They then use these calibrations to document the contribution of the
differences between these equilibria to cross-country income differences. Their model assumes
that two different productive sectors co-exist within a country, a traditional sector (agriculture)
producing under diminishing returns to all its factors, and a modern sector (non-agriculture) that
exhibits increasing returns that are external to the firm. They find that, depending on the degree of
increasing returns assumed for the non-agricultural sector, their model can account from between
15% and 60% of the observed inequality in income distribution across countries. The model
however cannot account for the large differences in income observed between poor and rich
countries. For extreme values of the increasing returns, the model predicts that output in the high
equilibrium is about 3 times output in the low equilibrium, far below the fifty fold difference
observed between the poorest and richest countries in the world. The model therefore can at most
account for the difference between poor and middle income countries. The elegance and
simplicity of the model, which allows it to determine the underlying alternative equilibrium for
each country in the world with minor data requirements, comes at two costs. First, in the simplest
version of the model the low equilibrium is unstable, so the authors have to resort to some ad-hoc
assumptions about the presence of a fixed cost of switching sectors. Second, the model is static
and focuses on the comparison of the different possible steady states of a country. So, the model
is mute about how the equilibrium is selected for each country, and, most importantly, it does not
allow us to analyze the dynamics of a country's income, the impact of policy interventions, or the
size or type of policies required to move a country away from its low equilibrium.
The second paper that adopts a calibration approach is Caucutt and Kumar (2004). They
calibrate models with multiple equilibria that capture four main mechanisms: coordination
failures, inappropriate mix of occupations, insufficient human capital accumulation, and political-
economy considerations. They argue that while it is possible to find plausible parameter values
that deliver multiple equilibria in these models, the existence of this multiplicity is also quite
sensitive to small variations in these parameters. Based on this they caution against policy
prescriptions for one-time interventions designed to jump countries from a bad equilibrium to a
good one. Although both the methodological approach and the conclusions of this paper are quite
similar to ours, our paper differs from theirs in two respects. First, we focus on different
mechanisms generating poverty traps than they do. Second, and perhaps more important, Caucutt
and Kumar (2004) are primarily interested in understanding whether necessary conditions for the
7
existence of multiple equilibria are satisfied for plausible parameter values. They do not ask, as
we do, whether the magnitudes of the income differences attributable to these multiple equilibria
are quantitatively reasonable.
3. Saving-Based Models of Poverty Traps
In this section of the paper we discuss one of the most popular models of poverty traps in
which the source of the trap is inadequate saving at low levels of development. In these models,
aid which augments domestic saving and finances accumulation can play a role in freeing
countries from poverty traps. We first illustrate the basic mechanism using a very simple Solow
growth model in which saving increases exogenously with the level of development. We next
take seriously the main underlying theoretical mechanism why saving rates might increase with
income: the influence of subsistence consumption. We present a Ramsey growth model with
subsistence consumption and show that, while it does not have a stable low-level equilibrium like
the poverty trap in the Solow model, it can exhibit poverty-trap-like features such as persistent
slow growth for very long periods of time. Besides being usually cited as an explanation for
poverty traps, the subsistence consumption model is representative of a class of models where
multiple equilibria result only from the form of the preferences, and we speculate that the
conclusions obtained in this case extend to other models of this class. Finally, we go to the data
and ask whether there is any evidence that (a) saving rates exhibit the sort of nonlinear
relationship with income required for the existence of a poverty trap, and (b) the historical
relationship between aid, saving, and growth is consistent with escapes from poverty traps.
The Basic Mechanism
We begin by using the Solow growth model to illustrate a saving-based poverty trap and
to perform some simple calibration exercises. With a Cobb-Douglas technology the familiar
dynamics of the capital stock per worker in the Solow model are given by:
(1) k = s(k) Ak - (n + )k
where k is the capital stock per capita, A is the exogenous level of technology, is the output
elasticity of capital, and n and are population growth and depreciation rates. The key feature of
the Solow model is its exogenous saving rate. This is an important simplification which we will
8
relax shortly. However, this assumption is very useful because it allows us to simply illustrate the
existence of a saving-based poverty trap. In particular we allow the saving rate to be an
exogenous function of the capital stock per capita, s(k), and assume that the saving rate is
~
constant at some low rate until a threshold value of the capital stock, k , is reached, and then it
jumps to a constant higher rate, i.e.
k
(2) s(k) = ssH ,, kk
L
> k
Figure 1 shows the familiar Solow diagram for this economy. There are two steady
states, indicated by kL and kH. The lower one can be thought of as a poverty trap, since if a
country starts out below kL, it will grow until it reaches the low steady state, and then remains
there forever. On the other hand, if a country starts out between k and kH , and therefore has a
high saving rate, it will grow steadily until it reaches the high steady state. The basic intuition for
the poverty trap is simple: at low levels of saving, investment is so low that it can only sustain a
very small capital stock per capita.
We have calibrated Figure 1 so that it captures what we think are some key features of
reality. We have drawn the low equilibrium at a capital stock per capita of $1500, which is
approximately equal to the population-weighted average capital stock per capita in Sub-Saharan
Africa. This is also roughly equal to output per capita in the region, implying a capital-output
ratio of around one. The capital-output ratio in the low steady state is given by sL /(n + ) . For a
depreciation rate of 6% and population growth rate of 4%, this implies that the saving rate in the
low steady state must be 10%, which is actually just a bit higher than the saving rate of 8% for
Sub-Saharan Africa as a whole.3 We then choose the level of technology, A, to ensure that the
steady state level of the capital stock at the low equilibrium, i.e. kL = sL A 1/(1- )
n + is equal to
$1500. For a benchmark value of =0.5, this implies that A=1.22. We have drawn the high
steady state under the assumption that saving rates are twice that in the low steady state, but that
there are no other changes in any of the parameters of the model. Since the ratio of the capital
3Note that all calculations are done using PPP-adjusted data. Since the relative price of investment
goods tends to be higher in poor countries than in rich countries, saving and investment rates are
substantially lower when evaluated at PPP than in local currency terms.
9
kH
stocks in the high and low steady states is = H , for our benchmark value of this
kL ssL 1/(1- )
implies that the capital stock in the high steady state is four times that in the low steady state, at
kH =$6000.
Although Figure 1 shows that we can generate a growth model with a poverty trap that
roughly matches Africa's experience, it is important to note that the low poverty trap equilibrium
exists only by assumption. Suppose for example that the point at which saving jumps to the
higher rate occurs to the left of what is now shown as the poverty trap equilibrium. In this case,
there would be no poverty trap, and a country starting out at the low capital stock would steadily
grow until it reaches the high steady state. Similarly, if population growth or depreciation were
slightly lower, the low equilibrium can also disappear as the diagonal line rotates downwards,
leaving only a single stable high equilibrium. Moreover, if we were to introduce exogenous
technical progress into the model, then the poverty trap equilibrium would also disappear,
provided that the threshold level of capital per capita k is fixed in absolute terms. If this is the
case, thanks to exogenous technical progress all countries will eventually cross this threshold and
the poverty trap vanishes as all countries will have the high saving rates.
Suppose despite all this that the poverty trap exists, and that some countries find
themselves stuck in it. How can aid help to break a country free of the poverty trap shown here?
In the model, the reason countries are in a poverty trap is because their saving (and hence
investment) rates are low. The role for aid in such a model is to augment saving sufficiently to
allow the country to accumulate capital and grow to the point where the domestic saving rate
jumps to the higher level. After the saving rate has increased, no further aid is required and the
country grows steadily until it reaches the high equilibrium. To take a specific numerical
example, suppose that we assume that the country in the low equilibrium receives a fixed fraction
of its GDP in aid for a ten year period, and assume further that all of this aid is used to finance
capital accumulation.4 We can then calculate how large the aid inflow must be to ensure that the
country just reaches a per capita capital stock of k at the end of ten years. After this point, the
country will embark on sustained growth and eventually reach the high steady state without
requiring further aid.
4 A ten-year period can be motivated by the time between now and 2015 when the Millenium
Development Goals are supposed to be attained.
10
The results of this exercise are reported in Table 1. Clearly, the amount of aid required
depends on how far the threshold level of capital, k , is from the capital stock in the poverty trap
equilibrium. The columns of the table correspond to different values of k , and the rows
correspond to different assumptions on the degree of diminishing returns, . For the benchmark
value of =0.5, aid inflows ranging from 4 percent to 25 percent of GDP would be required for a
ten-year period in order to break the country out of the poverty trap. Obviously, the higher the
threshold level of capital at which saving rates increase, the greater is the aid required. In
addition, the stronger are diminishing returns, i.e. the lower is , the more aid is required.
This very simple example illustrates what we think are some of the main messages of the
paper. First, while it is clearly possible for poverty traps to exist, the mere fact that saving
increases with income is not sufficient for the existence of a poverty trap ­ it needs to do so in
just the right way. As we will see shortly cross-country evidence is not very compelling that the
required nonlinearities in saving rates exist. Second, even if a poverty trap does exist, the size of
the policy intervention required to get countries out of poverty traps is very sensitive to the
parameters of the specific model of poverty traps, in this case, the precise point at which saving
rates jump, as well as the extent of diminishing returns, .
Subsistence Consumption
An important limitation of the example above is that we simply assumed that saving rates
increase at some level of development. In a fully specified model, saving rates are endogenously
determined by optimizing agents. The question therefore is what types of preferences can
generate saving functions whose behavior is consistent with the existence of poverty traps. One
important explanation offered in the literature is based on the existence of subsistence
consumption (see Azariadis (1996), Ben-David (1998), Sachs (2004)). The idea behind this
explanation is that poor households do not save because they have to use their income to meet
basic consumptions needs, but once these needs are met households may save quite a lot of their
income. We now present and calibrate a model that formalizes this intuition by explicitly
introducing a minimum consumption level in the preferences of the representative consumer in an
otherwise standard Ramsey model and deriving the optimal consumption and saving behavior
11
under these conditions.5 The model is closely related to Ben-David (1998), which analyzes the
implications of a model with subsistence consumption under exogenous growth.
The representative household chooses a path for consumption to maximize its inter-
temporal utility:
(3) U = u(ct)e-tdt
0
where c is the consumption per member of the household and the utility function is:
-
(4) u(c) = (c -c)1 -1
1-
The minimum subsistence consumption level is captured by c . The representative
household is endowed with a constant returns production technology F(K, L) and with an
endowment of labor that grows exogenously at rate n . Capital depreciates at a constant rate .
Under these conditions, the evolution of the capital per worker is given by
i
(5) k = f (k) -c - (n + )k ,
where k denotes capital per worker, and f (k) = F(K / L,1) represents the production function
in intensive form. The problem of the representative household is therefore
max U = u(ct)e-tdt
c 0
(6) s.t.
i
k = f (k) -c - (n + )k
which is the standard Ramsey problem.
The first-order conditions of this problem are given by the following two differential
equations:
5This is of course not the only way to generate a saving rate that increases with the level of
development. For example, Chakraborty (2004) develops a two-period overlapping generations growth
model in which the probability of survival to the second period is endogenously determined as a function of
public health care expenditures. The model exhibits an (unstable) low-level equilibrium in which the capital
stock, income, and health spending are low. This in turn means that the probability of survival is low, and
individuals rationally respond to this by saving less for their old age.
12
i
c = (c - c) (f '(k) - n - - )
(7)
i
k = f (k) -c -(n + )k
The equations show that in contrast to the standard Ramsey model, c = c is also a stable
locus for the consumption dynamics. The implications can be seen in the phase diagram depicted
in Figure 2. Compared with the standard Ramsey model, this model exhibits an additional
unstable equilibrium at the point (c,kl ) , and it also has the property that for capital stocks below
kl the high equilibrium ( k*,c*) cannot be attained. So, an economy that starts at (c,kl) can
remain indefinitely at the subsistence level, and an economy that starts with capital below kl will
maintain its consumption at the subsistence level and keep depleting its capital stock until
eventually capital is exhausted and the economy ends up at zero consumption. Of course, even for
the cases in which the high equilibrium of the standard Ramsey model is attained, the transitional
dynamics of this model will differ substantially from those of the case with no subsistence
consumption.
It is important to notice that this model is an example of a class of models where the
multiplicity of equilibria results only from the form of the preferences. In general, the first order
condition for the evolution of consumption in a Ramsey model corresponds to
c = (c)( f '(k) - n - - ), where (c) is the inverse of the intertemporal elasticity of
i
substitution. It is clear from this equation, that, unless (c) = 0 , there is only one possible
equilibrium for the level of capital per worker, given by f'(k)=n++. So, the only way to obtain
multiple equilibria based only on the preferences is that they satisfy the condition (c) = 0 for
some positive value of c. Besides the Stone-Geary preferences, we are not aware of other
standard specifications of the utility function that deliver this condition. We speculate therefore
that the conclusions from this model apply to a much more general class of preference-based
explanations for multiple equililibria. An important distinction between this model and the Solow
model of the previous section is that the low equilibrium is unstable. As a result, countries cannot
be "trapped" at this low equilibrium unless by some remarkable coincidence they begin just at the
13
low equilibrium.6 In other words, the model does not deliver a poverty trap. The reason is that,
although the saving rate in the model is initially increasing with income, it does not have the s-
shape (as a function of income) that is necessary to obtain multiple stable equilibria.
The failure of this model to generate a "standard" poverty trap shows how difficult it is to
obtain such a trap as a result of optimizing behavior without assuming functional forms that are
dangerously close to simply assuming that poverty traps exist. For example, Azariadis (1996)
presents a OLG model of subsistence consumption similar to ours that delivers a poverty trap by
assuming that the future level of subsistence consumption a step function of current wealth, hence
making richer households effectively more patient. So, the way this model generates a poverty
trap is by replacing the assumption of an s-shaped saving function with the assumption of an s-
shaped discount factor.
Although our model does not deliver a standard poverty trap, its transitional dynamics
may still generate something that looks like a poverty trap in the medium run. An economy that
starts very close to the unstable equilibrium can exhibit consumption close to subsistence, low
saving rates, and low growth for a significant length of time.7 In order to take this possibility
seriously, we need to calibrate the key parameters of the model. We will assume that the
production function is Cobb-Douglas f (k) = Ak , and specify values for the parameters
,,n,,and . For the benchmark specification we will set the share of capital = 0.5,
which is larger than the value of 0.36 typically used for developed countries (e.g. Kehoe and Perri
6 So, this model cannot explain long run differences between countries unless we are willing to
consider the zero consumption equilibrium as a possibility, or to assume that countries can remain
indefinitely at the unstable intermediate equilibrium. Both alternatives are unsatisfactory. The zero
consumption equilibrium is present in the standard Ramsey model, so it cannot be considered as the
specific manner in which subsistence consumption induces a poverty trap. The only difference with the
standard model in this respect is that in the model with subsistence consumption the zero equilibrium is the
only possible steady state for an economy starting from very low levels of capital, which makes this
equilibrium more likely. However, the model has the counterfactual implication that during the transition to
the zero equilibrium an economy would exhibit negative saving rates. Also, it is hard to provide an
economic intuition for an equilibrium with zero consumption in a model that explicitly includes subsistence
consumption levels. Regarding the second possibility, by definition an unstable equilibrium cannot account
for long run differences unless we are willing to assume the absence of any type of shocks.
7 A similar point was made by Steger (2000).
14
(2002)). For the robustness analysis we will vary the capital share between the value of 0.4
considered by Coe, Helpman, and Hoffmaister (1995), and 0.6.8 The sum of the depreciation rate
and population growth (n) was set at 0.1, and the rate of intertemporal preference at 0.05
which is consistent with the standard quarterly value of 0.99 for the discount factor used in the
RBC literature. Finally, the coefficient of risk-aversion (or the inverse of the intertemporal rate of
substitution) was set at 2, which is also standard.9 The parameter A was chosen to calibrate
the unstable low equilibrium to the level of consumption per capita and capital per-worker of the
poorest African country in 1996 (Democratic Republic of Congo with a consumption per capita
of about $300 and capital per worker of $330). This seems like a reasonable lower bound for a
subsistence level of consumption, and it is slightly smaller than the $300 in 1980 dollars
computed by Ben-David (1998) as the least cost requirement for sustaining an individual's
minimum dietary needs.
i i
The bottom panel of Figure 2 shows the c = 0 and the k = 0 locus in the (c, k) space for
the benchmark economy. It also shows the stable manifold of the economy, which was computed
numerically using the time-elimination method of Mulligan and Sala-i-Martin (1991). In this
parameterization, the levels of consumption and capital per person at the high stable equilibrium
correspond to $747 and $3,434 respectively. These levels are very low. So, a first quantitative
implication of the calibration exercise is that the high equilibrium consistent with the poorest
African country being at its subsistence level implies that, even after reaching the high
equilibrium, per capita consumption will still be below that of the median African country, and
well below that of middle income countries. Of course these magnitudes are sensitive to the
specific parameterization, but, as will be shown below, taking them to more reasonable levels
would require unlikely magnitudes for key parameters of the economy.
The effect of subsistence consumption on the pattern of growth for an economy that starts
at an initial level of capital 1 percent above the low equilibrium is shown in Figure 3. For
comparison, the figure also presents the evolution of capital per worker of an economy that starts
8 We cannot increase the value of further and simultaneously being able to calibrate the
unstable equilibrium of the economy with the values observed for low income countries keeping the rest of
the parameters fixed.
9Standard values for range between 1 and 3. As it will be pointed out below, the results are not
significantly affected by varying within this range.
15
with a much higher level of capital equal to the one observed for the median African economy,
and of a standard Ramsey economy with the same parameters. The top-left panel shows that the
convergence of the capital stock to the steady state is significantly slower for the benchmark
economy. The remaining panels show that the persistence of low levels of capital per worker for
the benchmark economy also translates in a low and persistent level of consumption, growth, and
saving rates. During the first 20 years, average consumption and capital per worker are $302 and
$339, which are barely above the subsistence levels, the saving rate averages 11 percent, and the
average growth rate is 0.5 percent. These are magnitudes that roughly fit the experience of poor
countries.10
The previous figures suggest that persistent low output and slow capital accumulation
observed in African countries could be explained by assuming that these countries are close to
their subsistence level of consumption. While these economies would not be "trapped" in the
usual sense because they would eventually reach the high equilibrium, they would nevertheless
display slow growth performance for an extended period of time. Over time, however, the
growth rate accelerates as subsistence consumption becomes less important, and finally the
growth rate declines as the economy approaches the stable high equilibrium.
However, it is important to note that this poverty trap-like behavior is very sensitive to
initial conditions. In particular, the length of time that low growth persists depends crucially on
how close the initial level of capital is to the level corresponding to the low unstable equilibrium
kl . Table 2 illustrates this point by reporting the time required to close half of the distance to the
steady state level of capital, assuming that the level of capital observed in low income countries is
ten, one, one-tenth, and one-hundredth percent above the low unstable equilibrium. For the
benchmark economy halving the distance to the steady state would take 54 years, as compared
with just 12 years for the no-subsistence-consumption economy. However, a country that starts
out 10 percent above the capital stock in the low equilibrium would take 41 years, and the median
(un-weighted) African economy with a capital stock of $934 per capita would take only 17 years.
In fact, it is apparent from the figures that, under the benchmark parameterization, the median
African country should behave very similarly to the standard Ramsey economy. In other words,
10Under this parameterization, the steady-state saving rate is about 33%. Although this magnitude
is high, it can be easily reduced by increasing the discount rate without significantly affecting the
properties of the solution. For example, in order to induce a steady-state saving rate of 25% we need to
double the discount rate to 0.1, which correspond to an underlying annual discount factor of about 0.9.
16
the existence of subsistence consumption should be practically irrelevant for the dynamics of the
median African country, and even more so for countries richer than the median.
Table 2 also shows that the main quantitative implications of the model for growth near
subsistence levels of consumption are robust to the capital share . We consider a range of
values for this parameter, and in each case re-calibrate the constant A to make the low unstable
equilibrium correspond to the values of consumption and capital observed for the poorest African
country.11 Although the level of the high equilibrium varies significantly, the pattern of
convergence looks similar in all three cases. In particular the times required to close half the gap
to the steady state are 59 years ( = 0.4), 54 years ( = 0.5), and 56 years ( = 0.6). Comparing
these differences with those obtained for economies that start at different distances from the low
unstable equilibrium makes clear that the main determinant of the speed of convergence is the
position of the initial conditions with respect to kl .
In summary, the results from these simulations suggest that the model with subsistence
consumption has the potential to generate something very similar to a poverty trap if we are
willing to assume that poor African countries are very close to subsistence. However, the flipside
of this assumption is that the median African country (and those above the median) should have
very high saving rates and be close to its high long run equilibrium. One possible way of
reconciling the predictions of the model with the data would be to assume that the subsistence
consumption levels are country specific, so that most low income African countries are close to
their own subsistence levels, but this would require us to argue that the subsistence consumption
level can vary significantly across countries. The consumption of the poorest African country is
one-third of the consumption of the median African country (PPP adjusted), and one-ninth of the
consumption of the richest African country. Analogously, assuming a common but higher level of
subsistence consumption, for example the consumption of the median African country, implies
that half of the countries below that level would be drifting towards the zero consumption
equilibrium, which is equally unappealing. These quantitative implications of the model suggest
that the subsistence consumption story as a source of poverty traps has serious problems matching
the data. Nevertheless, in what follows we will maintain the assumption that the persistence of
11Notice that because of the calibration of the constant A the effect of on the steady state is in
principle ambiguous. In practice, however, the standard increasing effect dominates.
17
low levels of capital and consumption in poor countries results from the fact that they are close to
the subsistence level, and ask how these countries can accelerate their escape from this situation.
What is the role of aid in this economy? As in the previous example, aid that augments
saving and domestic capital accumulation can bring the country toward the high steady state.
Unlike in the previous example, however, in the absence of aid countries would eventually get to
the high equilibrium on their own: all aid can do is accelerating this process. To get a sense of
the magnitude of the interventions involved, we compute the annual aid transfer as a fraction of
GDP necessary to double and quadruple the initial level of capital in 10 years.12 Since saving
rates are determined endogenously, we also need to make assumptions on how aid is delivered.
We consider two cases: (i) all transfers are exogenously directed to capital accumulation; (ii) the
allocation of transfers is left to the recipient country, but they are perceived as permanent. These
two cases give us a lower and upper bound for the more realistic case in which the transfers have
some bias towards capital accumulation and are at least partially perceived as transitory.
The values obtained for the transfers are reported in Table 3 for the benchmark economy,
the median African country, the Ramsey case, and for alternative values for the capital share
( a = 0.4,0.6 ). For the benchmark economy, the annual transfer, as a fraction of GDP, required to
double the capital per worker in a 10 year period is about 4 percent in the case in which all the
transfer goes to capital accumulation and 9 percent if the transfer goes to income but is perceived
as permanent (see columns (4) and (5)). As mentioned above, these two values represent a lower
and upper bound for the actual transfers required assuming different degrees of anticipation of the
end of the policy and biases towards capital accumulation. For this economy, doubling the initial
capital stock would have taken almost 30 years if left to itself (see column (6)). Doubling the
capital stock in 10 years therefore takes approximately 20 years from the development process of
the baseline economy. Quadrupling the capital stock however requires a much larger effort, with
transfers somewhere between 13 and 33 percent of GDP for 10 years.
Figure 4 shows the effects of aid on saving and growth in the first scenario of doubling
the per capita capital stock. Remember that, in each graph, the two exercises represent a lower
and upper bound for the true effects that should be observed. If we assume that the transfers are
12 For a very poor country with per capita income of $300 and a Gini coefficient of 40 to halve the
headcount measure of poverty with a 1$ per day poverty line, income needs to roughly double (assuming
an unchanged lognormal distribution of income). This in turn requires a quadrupling of the capital stock for
our benchmark case of =0.5.
18
fully invested, growth rates are higher on impact. Interestingly, for the case in which the transfers
are perceived as permanent, we observe an initial decline in the growth rates. A permanent
transfer of a fraction of GDP is in the model akin to an increase in TFP, which makes households
consider that their old saving plan was excessive and review it by increasing consumption.
Going back to Table 3, we observe that the transfers required to double the capital stock
for the median African country are higher than those obtained for the benchmark economy
because it is closer to the steady state, and, therefore, diminishing returns are more important for
the median country than for the country that is close to subsistence. For the median African
country, quadrupling the capital would take it above the steady state level, so the exercise is
omitted. The last two rows show that the transfers are decreasing in . For doubling the capital,
the lower bounds are about 5.4 percent for the low economy and 2.3 percent for the high one.
The corresponding upper bounds are 15.5 and 6.2 percent respectively. In general, the bounds are
not very sensitive to reasonable changes in the elasticity of substitution (not reported).
There is a striking difference in the magnitude of required aid inflows in this example
compared with the previous example based on the Solow model. To see this note that in Table 1
the case of attaining a threshold capital stock of $3000 in ten years corresponds to a doubling of
the initial capital stock. In the Solow economy doubling the capital stock required 10.5 percent of
GDP in aid for 10 years for the benchmark value of =0.5. In contrast, in Table 3 we see that
doubling the capital stock in the Ramsey economy with subsistence consumption requires only
3.9 percent of GDP in aid for the same capital share when aid goes directly to investment. The
reason for this difference is that in the Solow economy, domestic saving by assumption remains
constant at 10 percent of GDP and does not respond to the aid inflow. In contrast, in the
subsistence economy model there is a very strong increase in saving. In the absence of aid saving
rates would have averaged 10% of GDP in the first 10 years, while with aid saving rates increase
sharply and average 16% of GDP during this period. This additional aid-induced domestic saving
and investment has a strong effect on growth over and above the direct effects of aid.
In summary, the model could be used to rationalize the persistence of low levels of
consumption and capital observed in low income African countries by assuming that they are
close to the subsistence level, but this requires either to assume that there is significant variation
in the levels of subsistence consumption across African countries or that half of these countries
should exhibit high saving and growth rates. Also, taking the model seriously would imply that
relatively modest transfers, well within the magnitudes we observe historically should have an
19
important effect on the performance of these countries. We next ask whether this has been the
case empirically.
Empirical Assessment
In this sub-section we assess the empirical relevance of the saving based poverty trap
models discussed above. We first ask whether the cross-sectional and time-series behaviour of
saving is consistent with models of poverty traps. We then ask whether there is any evidence that
in the past aid has helped countries to escape from saving-based poverty traps.
Consider first the nonlinear relationship between saving and the level of development that
is required for a poverty trap to exist in the Solow model. In the left-hand panels of Figure 5 we
plot the empirical counterpart of this relationship. In each panel we have gross domestic
investment rates on the vertical axis, and on the horizontal axis we have the capital stock.13 The
top panel plots data for all countries, the second only for low-income countries, and the third only
for countries in Sub-Saharan Africa. In all these graphs we have fitted a third-order polynomial
relationship to pick up any non-linearities that might be present in the data. These non-linearities
do appear to be present in the cross-sectional relationship between investment rates and the level
of development. In all three cases the patterns are broadly similar, with sharp increases in saving
at very low capital stocks, then a fairly flat or even declining section, and then finally again an
increasing section. Note however that these turning points occur at very different levels of the
capital stock per capita, as the scale for the top graph is very different from the bottom two
panels. In particular, for the low-income and Sub-Saharan Africa samples the turning points all
occur at capital stocks less than $1500 per capita. Interestingly also, some of these nonlinearities
are statistically significant: the quadratic and cubic terms in the first and third panels are
statistically significant at the 5 percent level. At the same time, however, a visual inspection of
the three graphs suggests that the shape of these nonlinearities is likely to be quite sensitive to
extreme observations.
13 Throughout the paper we use data on investment rates as a proxy for saving rates. We do this
mainly for reasons of data availability, as gross domestic investment rates are available for many more
country-year observations than is gross national saving, reflecting gaps in data on current accounts. While
many poor countries of course receive substantial recorded capital inflows, many also have substantial
unrecorded capital outflows and so it is not obvious a priori the extent to which our findings are affected by
this approximation.
20
Despite the presence and significance of these nonlinearities, the patterns shown in the
left side of Figure 5 are not very promising for the poverty trap story. Recall that for the poverty
trap story to work, we need to find low saving rates at low levels of development, and then a
sharp increase over some intermediate range before leveling out at a high rate. This is just the
opposite of what we see in the data. The consequences of these saving patterns for the Solow
model are shown in the right panel of Figure 5. In the top panel the fact that saving rates are
increasing and concave over the relevant range implies a unique stable equilibrium around a
capital stock of $2000 per capita. For the low-income country sample we find a stable equilibrium
around $4000 per capita and two unstable ones around $400 and $7000. For the Sub-Saharan
Africa sample there is an unstable low equilibrium around $1000 and a stable high equilibrium at
$6000. In none of these cases do we find a sufficiently S-shaped pattern in saving rates to
generate a stable low-level equilibrium that might be thought of as a poverty trap, together with a
stable high equilibrium that countries might be able to attain with appropriately large amounts of
aid.
The bottom line here is that the cross-sectional patterns in saving rates do not appear to
be particularly supportive of the existence of poverty traps. This is not because saving rates do
not increase with the level of development. Rather the data suggest that they do not increase in
the right nonlinear fashion required for the existence of a poverty trap.
We now turn to the empirical predictions of the Ramsey economy with subsistence
consumption. A key prediction of this model, which can readily be seen from Figure 3, is that
countries starting out near subsistence levels will eventually experience sharp increases in saving
rates and sharp accelerations in growth as they reach a level of development where subsistence
considerations are no longer important.
For the baseline calibrations a country starting off near subsistence will see its saving
rates roughly double to 10 percent to 20 percent during the first 30 years, and growth will
accelerate from near zero to 5 percent per year.14 A first question to ask of the data is whether
jumps in saving rates of this magnitude are ever observed. In the top panel of Figure 6 we graph
the change in decadal average investment rates between 1960 and 1990 on the vertical axis, and
the initial log-level of per capita GDP in 1960 on the horizontal axis.
14 The jump in saving would be large even if we had a higher discount rate and therefore a lower
steady-state saving rate. For example, in the case in which we double the discount rate, which induces a
steady-state saving rate of 25%, savings would still increase from 10% to 17% in 10 years.
21
The first thing to notice from this graph is that such large increases in saving rates, even
over a 30-year period, are very unusual. Only three countries (Korea, Equatorial Guinea, and
Lesotho) have had such dramatic increases in investment rates.15 Even if we consider more
modest increases of 10 percent of GDP, we find only six more countries with increases in saving
in the 10-20 percent range. This suggests that accelerations in saving of the magnitude suggested
by subsistence consumption models are rare. Second, there is no obvious pattern in the data
suggesting that accelerations in saving are more likely at low income levels. There is only a small
and insignificant negative correlation between changes in saving and initial income. Moreover the
large increases in saving above 10 percent of GDP are fairly evenly distributed among countries
with per capita incomes ranging from about $600 to $3000 in 1960. In contrast, the benchmark
calibration of the model implies that large increases in saving rates occur in a range much closer
to subsistence levels. In fact, according to the calibration, the saving rate reaches 15% at about
$424 of capital per capita (which according to the other parameters implies a level of income per
capita of about $350). In the bottom panel of Figure 6 we ask whether large increases in saving
tend to be accompanied by growth accelerations, as the model with poverty traps would suggest.
There is a small and statistically insignificant relationship between changes in investment rates
and changes in growth rates, contrary to the implication of the theory that saving and growth
accelerations should go together.
Finally, we have seen that even fairly modest aid inflows can have very strong effects on
growth in the subsistence economy model. This reflected both the direct effects of aid on capital
accumulation, as well as the indirect effect of sharp increases in saving as aid makes the
subsistence constraint less binding. In particular, the model predicts that aid should increase
investment substantially, and that this in turn leads to a sharp increase in growth. Depending on
the specific assumptions on the use of aid flows (whether they go completely to investment or to
general expenditure) the increase in investment can be even more than one by one. At the very
least, increases in investment should be roughly of the same magnitude than the aid flows (as a
percentage of GDP). We have already seen a rather weak relation between investment and growth
in Figure 6. In Figure 7 we explore the links between aid and investment. In the top panel we
plot the relationship between aid as a fraction of GDP and the investment rate, both averaged over
the 1990s. The first thing to note is that there have been cases of quite poor countries that have
15 In the case of Equatorial Guinea, large foreign investments in the oil sector are an important
factor, while in Lesotho migrant workers' remittances are an unusual contributing factor.
22
received very substantial aid inflows over 10-year periods, for example Guinea-Bissau at 17
percent of GDP, or Equatorial Guinea at 10 percent of GDP during the 1990s. Note also that these
aid ratios have GDP adjusted for purchasing power parity in the denominators. Had we used
market exchange rates in the denominator the aid ratios would be much higher.16 In short,
however, there have been quite a few cases of poor countries receiving aid flows on the order of
magnitude that our calibrations suggest would be required to get out of poverty traps.
Unfortunately, however, the poverty trap models also imply very large increases in
investment should occur with aid. Yet the simple correlation between aid ratios and investment
rates is weakly negative, both in levels (top panel) and in differences (bottom panel). The lack of
a significant relation between aid and investment survives the inclusion of controls for policy, and
holds among low-income countries and among Sub-Saharan African countries as well. A
particularly extreme example is the case of Sao Tome and Principe, which is suppressed in the
bottom panel because the observation is so extreme. Our data suggest that aid increased by 20%
of GDP between the 1980s and the 1990s, yet investment rates fell by 5%.
None of these simple stylized facts should be especially surprising. Rodrik (2000)
provides a careful analysis of episodes of "saving transitions" which he defines as increases of
five percent or more in saving rates. He finds only 20 such episodes during the period 1960-1995.
This suggests that escapes from poverty traps must be quite rare since their hallmark is a sharp
increase in saving rates. It is also interesting to note that the saving transitions he identifies occur
at a very wide range of per capita income levels, ranging from around $800 in China to over
$6000 in Mauritius. It seems difficult to interpret the saving transitions occurring at high income
levels as resulting from the escape from a poverty trap based on subsistence consumption simply
because these countries are so rich that subsistence considerations matter little. Finally, Rodrik
also documents that saving transitions do not tend to be accompanied by higher growth. Although
these sharp increases in saving tend to persist for long periods of time, growth accelerates only
for a relatively short period and quickly returns to its pre-transition level. This again seems
inconsistent with the poverty trap story.
16 Which of the two is more appropriate? This depends on whether aid dollars are used to buy
goods and services at international or at domestic prices. We are not aware of estimates of the division of
aid along these lines. Here we report the PPP aid ratios simply to be conservative about the magnitude of
aid ratios that we actually observe.
23
The other observation on the weak relationships between aid, investment and growth has
been amply documented in a series of recent papers. Easterly (1999) finds that in only six of 88
countries for which data are available does aid raise investment at least one by one, and in only
17 countries the effect is positive and statistically significant. He also finds virtually no evidence
that investment and growth are correlated within countries. For Africa in particular, Easterly and
Dollar (1999) find that in none of 34 African countries aid increases savings at least one by one.
Similarly, Devarajan, Easterly and Pack (2001, 2003) document that the effect of public
investment on growth is not significantly different from zero, and that the effect of private
investment on growth is significant only if Botswana is included in the sample. Based on this as
well as a variety of micro evidence they conclude that the productivity of investment in Africa is
extremely low.
In summary, while models of saving based poverty traps are quite intuitive and are
frequently invoked as a motivation for aid, we fail to find any clear evidence in support of them.
We have seen that there is little evidence in support of the nonlinearities in saving rates across
countries required to generate poverty traps. We have also seen that there are very few instances
of saving and growth accelerations that would be consistent with calibrated models of poverty
traps. And finally, as many others have noted, there appears to be very little evidence that aid has
an at least proportional effect on investment as these models would suggest. This is the case
despite the fact that there have been quite a few episodes of countries receiving very large aid
inflows that our calibrations suggest would be sufficient to break these countries out of poverty
traps. Overall this evidence leads us to doubt the empirical relevance of saving-based models of
poverty traps.
4. Technology-Based Models of Poverty Traps
We now turn to another popular argument for the existence of poverty traps: the idea that
there are sharp increases in productivity once a certain threshold level of development is attained.
This idea can be generated in a variety of models in which indivisibilities in production,
externalities across firms, public infrastructure, and many other factors play a role. In this section
we will not restrict ourselves to a particular mechanism generating the acceleration in
productivity. Rather we will review some fairly general requirements for the existence of a
24
technology-based poverty trap, and then we will show that these requirements do not appear to
hold in cross-country and firm-level data.
Theoretical mechanisms
To be consistent with perfect competition, theoretical models usually assume that
increasing returns are external to the firm. As surveyed by Caballero and Lyons (1990), standard
explanations for these economies of scale that are external to the firm but internal to the industry
or country include advantages of within-industry specialization, agglomeration, indivisibilities
and public intermediate inputs such as roads or other infrastructure. Some papers along these
lines include Bryant (1983), Weil (1989) and Durlauf (1991) who introduce some form of
externality in the production process, and Diamond (1982), Howitt (1985), and Howitt and
McAfee (1988) who analyze models of search and matching with thick market externalities. Most
of these models, however, are highly stylized and not amenable to calibration. Nevertheless, as
mentioned above they share the common feature that the increasing returns are assumed to be
external to the firm, so that, in a model of homogeneous firms, the technology of a representative
firm j has the form:
(8) Yj = A(E)F(K j, Lj)
where Yj , K j , and Lj are the total output, capital, and labor of firm j, and A(E) , the scale
factor capturing total factor productivity, depends on a measure E of the average or aggregate
level of activity. This dependence is taken as given by the atomistic firm. As usual, the function
F() is assumed to have constant returns to scale.
The crucial aspect of the specification above is the dependence of the scale factor on a
measure of aggregate activity. Different models take different stances on what is the source of the
external economies, which will determine what is the relevant variable that affects A. We will
consider three possibilities. In the first specification, the scale factor depends on the average level
of capital as captured in the capital per worker. This approach was introduced by Azariadis and
Drazen (1990) in their model of threshold externalities, and, as noted by Barro and Sala-I-Martin
(1991), it has the advantage of eliminating the scale effects that appear in models like the ones
presented by Romer (1986) and Barro (1990). In the second specification, we assume that the
scale factor depends on the aggregate level of output. This approach is similar to the one taken by
Caballero and Lyons (1990, 1992, 1994), and Cooper and Haltiwanger (1996). Finally, we also
consider the case in which the scale parameter depends on the aggregate level of capital, in the
spirit of the literature on capital-embodied knowledge started by Romer (1986). For each of these
25
specifications, we explore the restrictions that the existence of a stable low equilibrium imposes
on the technological parameters and then we check whether these restrictions are actually borne
out by the data.
The simplest specification for the case in which the scale factor depends on the level of
capital per worker is the one considered by Azariadis and Drazen (1990), where there is a
threshold level of capital at which there is a "technological jump". That is
(9) A(k) = A A if k k
> A otherwise
The ability of a technology like this to account for the differences in income observed in low
income countries will of course depend on the threshold capital level k . In a Solow framework,
the analysis will be very similar to the one presented in Section 2. This is because, in the Solow
model, the role of jumps in saving and jumps in technology are more or less interchangeable. In
particular, all of the discussion in the first part of Section 2 can be re-interpreted as a story in
which there is an exogenous improvement in productivity once capital per worker reaches a
certain level. In this framework a poverty trap featuring low productivity can exist since for a
given saving rate, low productivity means that investment is insufficient to generate sustained
growth in the capital stock.
As we shall see shortly in the data, it seems difficult to find empirical evidence for such a
discrete jump in the level of technology. However, it is possible to have a poverty trap with a less
dramatic nonlinearity in technology. In fact, we can use the Ramsey economy of the previous
section to ask what degree of increasing returns would be necessary over some range of the
production function to obtain a stable low equilibrium corresponding to a poverty trap. Assume
that TFP (A) is a function of aggregate capital stock per capita (A(k)) but that firms do not
internalize the presence of the external effect in their production decisions. Under these
assumptions, the two equations that govern the dynamics of the Ramsey economy are:
i
c = (A(k) f '(k)-n - - )
1
(10) c
i
k = A(k) f (k) -c -(n + )k
As in the standard Ramsey case, the equilibrium levels of capital are pinned down by the
dynamics of consumption. In a steady state with no consumption growth the capital level has to
satisfy the condition A(k) f '(k) = n + + . As A(k) and f '(k) are a increasing and
26
decreasing function of k respectively it is easy to see that it is possible for this equation to have
multiple solutions, depending on the specific form of the A(k) function. Assuming, as is
reasonable, that limk 0 A(k) < and limk A(k) < , the only way of inducing multiple
equilibria is for A(k) to be significantly steeper than f '(k) in some range. The situation is
depicted in Figure 8. The exact condition is:
A'(k)k f ''(k)k
(10) > - ,
A(k) f '(k)
that is, there must be a range where the elasticity of A(k) is higher than the elasticity of f '(k) .
Assuming that f (k) = k , the condition translates into
A'(k)k
(11) (k) = > (1-).
A(k)
The intuition behind this condition is that the stronger the diminishing returns of the
production function f () , the larger has to be the scale effect on TFP to offset the tendency of the
aggregate returns to decrease. Assuming that takes values somewhere between 0.4 and 0.6, the
condition requires values for the elasticity of productivity with respect to the capital stock per
worker to be between 0.4 and 0.6.
The above discussion is not significantly affected by assuming that the scale factor
depends on aggregate output (A=A(Y)) or on aggregate capital (A=A(K)) instead of depending on
capital per worker. Replacing these alternative functional forms in the problem and maintaining
the assumption that the increasing returns are external to the firm, first-order conditions of the
problem are very similar. A little bit of algebra then shows that the existence of a stable low
equilibrium--of the kind consistent with a poverty trap story ­ imposes similar constraints on the
elasticity of the scale factor with respect to its corresponding argument. In order to have a low
equilibrium, a necessary condition is that the implied elasticity of TFP with respect to the relevant
variable (aggregate output or aggregate capital) has to be strictly larger than 1- over some
range. Again, assuming that the capital share in factor costs in African countries is between 0.4
and 0.6, the condition also requires the elasticity to lie between these two values.
Empirical Assessment
A natural place to begin looking for evidence of these scale effects on productivity is in
the large empirical literature on estimating returns to scale. In this literature, the degree of returns
27
to scale observed in an industry (firm) is obtained by estimating a relationship between the
growth rate of value added (or output) and the growth rates of the different inputs:
(12) Y^ = 0 + 1K^ + 2L^ +
where the hat denotes the percentage change of a variable. The sum of the coefficients 1 and
2 denotes the degree of returns to scale of the production function, and this sum is usually the
value reported for the papers that empirically estimate the degree of returns to scale.17
Under our assumed functional form for the production function, the growth rate of
aggregate output is given by
(13) Y^ =K^ + (1-)L^ + A^
where is the share of capital on total factor costs. For the cases in which A depends on
aggregate output and aggregate capital, the growth rate of the scale factor can be expressed as a
function of its elasticity and the growth rate of the underlying variable, so that the expression
for the growth rate of output corresponds to
K^ + 1-
(14) Y^ = L^
1- 1-
in the case in which the scale factor is a function of aggregate output, and to
(15) Y^ =K^ + (1-)L^ + K^
in the case in which the scale factor is a function of aggregate capital.
These two expressions allow us to compare the magnitude of the returns to scale
estimated in the literature with those required for the existence of technology-based poverty
traps. Recall that for a poverty trap to exist we require >1- over some range. For the case in
which the scale factor depends on total output, equation (14) tells us that the sum of the
estimated coefficients on capital and labor equals (1- )-1, which is increasing in . Assuming
17This estimation is not as simple as it seems. There are many technical complications in the
estimation of the parameters because of the endogeneity of the input choice, and also because of the
different implications of the aggregation process in the presence of increasing returns and monopoly power.
The interested reader is referred to the discussion in Basu and Fernald (1995).
28
that the capital share ranges from =0.4 to =0.6, this implies that estimates of returns to scale
must range from 1.7 to 2.5 in order for a poverty trap to exist. When the scale factor depends on
total capital, Equation (15) tells us that the sum of the estimated coefficients on capital and labor
is equal to 1+ . This in turn means that estimated returns to scale must range from 1.4 to 1.6
over some range of the production function.
These required levels of returns to scale ranging from 1.4 to 2.5 are much higher than
what is typically found in the literature. We know of only one study that approaches the lower
end of this range. Levinsohn and Petrin (1999) look at data from Chilean plants, and estimate
scale returns from value-added production functions that are above 1.20, and in some cases reach
up to 1.44. However, most other available studies only find constant to moderate increasing
returns. Using Mexican plant-level data, Tybout and West-brook (1995) obtained almost no
evidence of scale for large plants. For rich countries, Harrigan (1999) found no evidence of scale
in cross-country regressions. Many other OECD country studies point clearly to the existence of
moderate scale returns. Paul and Siegel (1999) estimated industry-level scale returns in the range
of 1.30 for many U.S. manufacturing industries, and Fuss and Gupta (1981) and Griliches and
Ringstad (1971) also estimate moderately increasing returns to scale for Canada and Norway
respectively.
The absence of empirical evidence of strong increasing returns casts doubt on these
technology-based explanations for poverty traps. However, it is important to note that
technology-based poverty traps could exist even in the absence of increasing returns in the
aggregate production function. In the first example we discussed, we assumed that total factor
productivity was a function of capital per capita. In this case, the aggregate production function
has constant returns to scale, but a poverty trap could nevertheless exist. Moreover, in this case
existing evidence on the degree of returns to scale is not helpful in assessing the empirical
relevance of this particular mechanism. In the remainder of this section, we directly examine the
relationship between estimates of total factor productivity and capital per worker using firm-level
and cross-country data.
The three panels of Figure 9 show the relation between (log) TFP and (log) capital per
worker, total capital, and output across countries. Total factor productivity is obtained as residuals
29
from a Cobb-Douglas production function in capital and labour, with a capital share of 0.5.18
Each panel also shows the estimated relation between the variables assuming both a simple linear
specification and a cubic polynomial to allow for the nonlinearities required to generate multiple
equilibria. In the first panel we observe a linear relation between (log) TFP and (log) capital per
worker. It is apparent that the polynomial specification does not offer a significant improvement
on the fit with respect to the linear specification. Overall, the estimated relationship suggests a
constant elasticity of about 0.2. Both the finding of a constant elasticity and its magnitude are
inconsistent with the requirements of multiple equilibria. A constant elasticity implies that the
function Af '(k) is either always decreasing or always increasing, which implies a unique
equilibrium (or none). Also, as determined above, the lower bound for (obtained by assuming
a very large capital share) is 0.4, which is almost twice the estimated value. The relation of TFP
with total capital, shown in the second panel, does not fare better. In fact, the figure shows
practically no relation between the two variables. Even if we take the econometric results at face
value, they also suggest a constant and small elasticity (about 0.06). As explained above, a
constant elasticity is inconsistent with multiple equilibria, and the implied value of the elasticity
is way below the magnitude required for multiplicity. Finally, the bottom panel shows the relation
of TFP with total output. It is apparent that the situation is almost identical to that obtained for
total capital.
The panels of Figure 10 repeat the exercise performed in Figure 9, but this time using
data from Colombian plants.19 The graphs plot the relations between the log of TFP at the plant
level and the log of the different measures of activity at the industry level after partialling-out
plant, industry, and year fixed effects. The conclusion that emerges from the three figures is
similar, and suggests that the scale factor is independent of these aggregate variables. The linear
specifications show a clearly flat relation between the log of firm level TFP and the aggregate
variables, and the non-linear specifications are undistinguishable from the linear ones. Again,
although these results should be taken with caution because of the caveats mentioned above, the
18 Results are similar using other values for the capital share. Note also that we make no effort to
adjust for the quality of labour input, as is often done in development accounting exercises. It could be that
the nonlinearity in productivity comes precisely from sharp improvements in worker skills at some level of
development, and so we would like our estimates of productivity to capture this.
19 The data on Colombian plants comes from Fernandes (2003). Industry level data was obtained
from UNIDO (2002).
30
message we want to emphasize is that the scant evidence available is far from supporting any
increasing or non-linear increasing returns external to the firm.
5. Concluding Remarks
There is by now an abundance of theoretical models of poverty traps capturing a wide
variety of self-reinforcing mechanisms through which poor countries might remain poor. What is
lacking is a similar body of empirical evidence in support of such models. This is particularly
unfortunate given the emphasis on poverty traps in recent discussions of aid policy. In this paper
we have tried to take seriously the quantitative implications of two leading mechanisms
generating poverty traps in aggregate growth models: low saving and low productivity at low
levels of development. We do not find much evidence in support of the idea that these two
mechanisms are empirically relevant. While saving rates and productivity do clearly increase
with income levels, we find that they do not increase in the right nonlinear way required to
generate poverty traps. We also find that a poverty-trap view of aid relying on these mechanisms
leads to counterfactual predictions for the relationship between aid, investment, and growth. This
leaves us skeptical of the popular idea that sufficiently large increases in aid will have
disproportionate effects on economic growth in low-income countries. This does not mean that
aid does not matter for growth, or for other development outcomes. Rather we do not find
evidence of threshold effects whereby sufficiently high levels of aid are necessary to "jump-start"
a sustainable growth process.
An important limitation of this paper is that we have only considered a few of the many
possible theoretical mechanisms generating poverty traps, and we have restricted ourselves to
highly aggregative models of growth. More empirical work on calibrating specific mechanism
using both micro and macro data may show that other self-reinforcing processes are empirically
important as explanations for underdevelopment. It may also be the case that a variety of
mechanisms operating simultaneously will deliver empirically relevant poverty traps. Results
along these lines may well provide a basis for the argument that large-scale increases in aid will
have disproportionate effects on growth. This work however remains to be done.
31
References
Azariadis, Costas and Allan Drazen, (1990), "Threshold Externalities in Economic
Development," The Quarterly Journal of Economics, 105(2), pages 501-26.
Azariadis, Costas, and John Stachurski, (2004), "Poverty Traps", Forthcoming in the Handbook
of Economic Growth, Aghion and Durlauf eds.
Azariadis, Costas, and John Stachurski, (2004b), "A Forward Projection of the Cross-Country
Income Distribution", Manuscript, UCLA.
Banerjee, Abhijit, (2001), "Contracting Constraints, Credit Markets, and Economic
Development", mimeo, MIT.
Banerjee, Abhijit, and Esther Duflo, (2004), "Growth Theory Trough the Lens of Development
Economics", mimeo, MIT.
Barro, Robert J, (1990), "Government Spending in a Simple Model of Endogenous Growth,"
Journal of Political Economy, 98(5), pages S103-26.
Bartelsman, Eric J, Ricardo J Caballero, and Richard K Lyons, (1994), "Customer- and Supplier-
Driven Externalities," American Economic Review, 84(4), pages 1075-84.
Basu, Susanto and John G. Fernald, (1995), "Are apparent productive spillovers a figment of
specification error?," Journal of Monetary Economics, 36(1), pages 165-188.
Ben-David, Dan, (1998), "Convergence Clubs and Subsistence Economies", Journal of
Development Economics, 55, 153-159.
Bloom, David E, Canning, David and Sevilla, Jaypee, (2003). "Geography and Poverty Traps,"
Journal of Economic Growth, vol. 8(4), pages 355-78, December.
Bryant, John, (1983), "A Simple Rational Expectations Keynes-Type Model," The Quarterly
Journal of Economics, 98 (3), pages 525-28.
32
Caballero, Ricardo J. and Richard K. Lyons, (1990), "Internal versus external economies in
European industry," European Economic Review, 34(4), pages 805-826.
Caucutt, Elizabeth and Krishna Kumar (2004). "Evaluating Explanations for Stagnation".
Manuscript, University of Western Ontario and USC Marshall School of Business.
Caballero, Ricardo J. and Richard K. Lyons, (1992), "External effects in U.S. procyclical
productivity," Journal of Monetary Economics, 29(2), pages 209-225.
Chakraborty, Shankha (2003). "Endogenous Lifetime and Economic Growth". Journal of
Economic Theory, forthcoming.
Cooper, Russell and John Haltiwanger, (1996), "Evidence on Macroeconomic
Complementarities," The Review of Economics and Statistics, 78(1), pages 78-93.
Devarajan, Shantayanan, William Easterly, and Howard Pack (2001). "Is Investment in Africa
Too High or Too Low? Macro- and Micro-Evidence". Journal of Africa Economies. 10:81-108.
Devarajan, Shantayanan, William Easterly, and Howard Pack (2003). "Low Investment is Not the
Constraint on African Development". Economic Development and Cultural Change.
Diamond, Peter A, (1982), "Aggregate Demand Management in Search Equilibrium," Journal of
Political Economy, 90(5), pages 881-94.
Dollar, David, and William Easterly (1999), "The Search for the Key: Aid, Investment, and
Policies in Africa", Journal of African Economies, 8 (4), 546-577.
Durlauf, Steven N, (1991), "Multiple Equilibria and Persistence in Aggregate Fluctuations,"
American Economic Review, 81(2), pages 70-74.
Easterly, William, (1999), "The ghost of financing gap: testing the growth model used in the
international financial institutions", Journal of Development Economics, 60, 423-438.
33
Fernandes, Ana Margarida, (2003), "Trade Policy, Trade Volumes, and Plant-Level Productivity
in Colombian Manufacturing Industries", World Bank Policy Research Working Paper 3064.
Feyrer, James (2003), "Convergence by Parts", Mimeo, Dartmouth College.
Fuss, Melvyn A. and V. K. Gupta (1981) "A Cost Function Approach to the Estimation of
Minimum Efficient Scale, Returns to Scale, and Suboptimal Capacity," European Economic
Review, 15 (2), 123­135.
Graham, Bryan S. and Jonathan Temple, 2004. "Rich Nations, Poor Nations: How much can
multiple equilibria explain?," mimeo, Harvard University.
Griliches, Zvi and Vidar Ringstad (1971) Economies of Scale and the Form of the Production
Function: An Econometric Study of Norwegian Manufacturing Establishment Data, Amsterdam:
North Holland.
Harrigan, James (1999) "Estimation of Cross-Country Differences in Industry Production
Functions," Journal of International Economics, 47 (2), 267­93.
Hausmann, Ricardo, Lant Pritchett, and Dani Rodrik (2004). "Growth Accelerations".
Manuscript, Kennedy School of Government.
Howitt, Peter, (1985), "Transaction Costs in the Theory of Unemployment," American Economic
Review, 75(1), pages 88-100.
Howitt, Peter and R Preston McAfee, (1988), "Stability of Equilibria with Externalities," The
Quarterly Journal of Economics, 103(2), pages 261-77.
Kehoe, Patrick J. and Fabrizio Perri, 2002, "International Business Cycles with Endogenous
Incomplete Markets." Econometrica, 70(3), pages 907-928.
Levinsohn, James and Amil Petrin (1999) "When Industries Become More Productive, Do Firms?
Investigating Productivity Dynamics." Mimeo, University of Michigan.
34
Lokshin, Michael and Martin Ravallion (2004). "Household Income Dynamics in Two
Transition Economies". Studies in Nonlinear Dynamics and Econometrics. 8(3), Article 4.
Mulligan, Casey, and Xavier Sala-i-Martin, (1991), "A Note on the Time Elimination Method for
Solving Recursive Dynamic Economic Models", NBER Technical Working Paper 116.
McKenzie, David and Christopher Woodruff (2004). "Is There and Empirical Basis for Poverty
Traps in Developing Countries?". Manuscript, Stanford and UCSD.
Paul, Catherine J. Morrison and Donald S. Siegel (1999) "Scale Economies and Industry
Agglomeration Externalities: A Dynamic Cost Function Approach," American Economic Review,
89 (2), 272­290.
Quah, Danny T, (1996). "Twin Peaks: Growth and Convergence in Models of Distribution
Dynamics," Economic Journal, 106 (127), pages 1045-55.
Quah, Danny T, (1993a), "Empirical Cross-Section Dynamics in Economic Growth". European
Economic Review 37, pp. 426­434.
Quah, Danny T, (1993b), "Galton's Fallacy and Tests of the Convergence Hypothesis". The
Scandinavian Journal of Economics 95, pp. 427­443.
Quah, Danny T, (1997), "Empirics for Growth and Distribution: Stratification, Polarization, and
Convergence Clubs". Journal of Economic Growth 2 (1997), pp. 27­59.
Rodrik, Dani (2000), "Saving Transitions". World Bank Economic Review. 14(3):481-507.
Romer, Paul M, (1986), "Increasing Returns and Long-run Growth," Journal of Political
Economy, 94(5), pages 1002-37.
Sachs, Jeffrey, John W. McArthur, Guido Schmidt-Traub, Margaret Kruk, Chandrika
Bahadur, Michael Faye, and Gordon McCord, (2004), "Ending Africa's Poverty Trap", Brooking
Papers on Economic Acitivity, 2004 (1).
35
Sachs, Jeffrey (2005). "Investing in Development: A Practical Plan to Achieve the Millenium
Development Goals. New York. UN Millenium Project.
Steger, Thomas, (2000), "Economic Growth with Subsistence Consumption", Jornal of
Development Economics, 62, 343-361.
Tybout, James R. and M. Daniel Westbrook (1995) "Trade Liberalization and the Dimensions of
Efficiency Change in Mexican Manufacturing Industries," Journal of International Economics,
39 (1-2), 53­78.
Weil, Philipe, (1989), "Increasing Returns and Animal Spirits", American Economic Review, 79,
889-894.
36
Table 1: Aid Required to Escape From Poverty Trap
Threshold Capital Stock ($000's)
2 3 4 5 6
0.25 4.6% 12.9% 20.6% 27.8% 34.7%
0.5 3.9% 10.5% 16.1% 21.0% 25.4%
0.75 3.4% 8.6% 12.6% 15.9% 18.7%
37
Table 2: Persistence of Slow Growth
Time to close half
Initial capital (%
the gap to ss
above low equilibrium)
(years)
10% 40
1% (benchmark) 54
0.10% 68
0.01% 85
Median African (300%) 24
Ramsey economy 15
1% (capital share = 0.4) 59
1% (capital share = 0.6) 56
38
Table 3: Aid and Growth with Subsistence Consumption
Annual Annual Annual
Annual transfer to transfer to Time to transfer to transfer to Time to
double initial double double quadruple quadruple quadruple
Parameterization
capital in 10 considering capital by initial capital considering capital by
periods (% GDP) permanent itself in 10 periods permanent itself
transfers (% GDP) transfers
(1) (2) (3) (4) (5) (6)
Baseline 0.039 0.093 29 0.1277 0.3252 42
Median african 0.048 0.128 15 -- -- --
Ramsey (c_=0) 0.000 0.000 3 0.000 0.000 8
a = 0.4 0.054 0.155 38 0.1742 0.511 69
a = 0.6 0.023 0.062 22 0.0871 0.1858 30
39
Figure 1: Solow Model With Poverty Trap
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
s x y
0.2
(n+d) x k
0.1
0
6kH
0 1 kL 2 3
~ 4 5 7 8
k Capital Stock Per Capita
40
Figure 2: Ramsey Economy with Subsistence Consumption
Phase Diagram
c*
c
kl k* kh
Calibrated Phase Diagram
1.5
0)
$00( 1
erk saddle path
or
wr dk/dt=0
pe
oni
pt c = csubs
ums 0.5
on
C
0
0 1 2 3 4 5 6 7 8
Capital per worker ($000)
41
Figure 3: Growth With Subsistence Consumption
Capital Stock Per Capita Consumption Per Capita
4
Baseline
0.8
3.5 Median african
Baseline
Ramsey
Median african
0.7
3 Ramsey
0) )
0.6
($00 2.5
er
rko 000$(rek
wr 2 0.5
wor
pelatipa erp
1.5 noi 0.4
pt
C um
1 0.3
Cons
0.5 0.2
0 0.1
0 20 40 60 80 100 120 140 160 180 200 0 20 40 60 80 100 120 140 160 180 200
Time (years) Time (years)
Growth Rate Saving Rate
0.35
Baseline 0.5
Median african Baseline
0.3
Ramsey 0.45 Median african
Ramsey
0.25 0.4
th
wor 0.2 0.35
Gl tear
taipa gn 0.3
0.15 via
C S
0.25
0.1
0.2
0.05
0.15
0 0.1
0 20 40 60 80 100 120 140 160 180 200 0 20 40 60 80 100 120 140 160 180 200
Time (years) Time (years)
42
Figure 4: Effects of Aid
Saving
0.35
Baseline
10 year transfer (1)
10 year transfer (2)
0.3
e 0.25
atr
ngi
av
S 0.2
0.15
0.1
0 20 40 60 80 100 120 140 160 180 200
Time (years)
Growth
0.1
Baseline
10 year transfer (1)
10 year transfer (2)
0.08
0.06
h
owtr
Glat 0.04
Capi
0.02
0
-0.02
0 20 40 60 80 100 120 140 160 180 200
Time (years)
43
Figure 5: Cross-Country Relation Between Saving and Capital Per Capita
Empirical Evidence Consequences for Solow Model
All Countries 1
70 0.9 (n+d) x k
60 0.8
0.7 s x y (All Countries)
50
Rate 0.6
40 0.5
30 0.4
0.3
vestmentnI 20
0.2
10 0.1
0 0
0 20 40 60 80 100 0 2 4 6 8 10
Capital Stock Per Capita Capital Stock Per Capita
Low-Income Countries 1
60 0.9 (n+d) x k
0.8
50
tea 0.7 s x y (LICS)
Rtne 40 0.6
0.5
30
0.4
20 0.3
vestm
In 10 0.2
0.1
0 0
0 5 10 15 0 2 4 6 8 10
Capital Stock Per Capita Capital Stock Per Capita
1
Sub-Saharan Africa
60 0.9 (n+d) x k
0.8
50
0.7 s x y (SSA)
Rate 40 0.6
0.5
30
0.4
20 0.3
vestmentnI10 0.2
0.1
0 0
0 5 10 15 0 2 4 6 8 10
Capital Stock Per Capita Capital Stock Per Capita
44
Figure 6: Changes in Saving and Growth, 1990s vs 1960s
Saving Accelerations
40
30
ate,
Rt 20
10
1960s
vestmennI vs 0
5 6 7 8 9 10
ni -10
eg 1990s
-20
anh y = -1.66x + 12.79
-30
C R2 = 0.02
-40
ln(GDP Per Capita in 1960)
Changes in Saving and Growth
0.15
et 0.1
Ra
ht 0.05
row
G 0
in -40 -20 0 20 40
-0.05
nge
Cha -0.1 y = 0.00x - 0.02
R2 = 0.04
-0.15
Change in Investment Rate
45
Figure 7: Aid and Investment
Cross-Section in 1990s
45
40 y = -0.27x + 14.10
P 35 R2 = 0.02
D
G/t 30
25
20
vestmennI15
10
5
0
0 10 20 30 40
Aid/GDP
Changes in 1980s vs 1990s
30
vs 25
y = -0.35x + 0.12
20 R2 = 0.02
1990s,t 15
10
5
0
vestmennI 1980s
-10 -5 -5 0 5 10
ni -10
eg -15
anh -20
C
-25
Change in Aid, 1990s vs 1980s
46
Figure 8: Technologically induced multiple equilibria
n + +
Af '(k)
k
kL kI kH
47
Figure 9: Cross-Country Relation Between TFP and Capital
Per Capita Capital
2 y = -0.0224x3 + 0.5807x2 - 4.6866x + 12.164
R2 = 0.6101
1.5
1
)
TFP 0.5
n(l
0
5 6 7 8 9 10 11 12
-0.5
y = 0.2259x - 1.3739
-1 R2 = 0.5762
ln(Capital Per Capita)
Total Capital
2
y = -0.0016x3 + 0.1243x2 - 3.0683x + 25.172
1.5 R2 = 0.1084
1
)
TFP 0.5
n(l
0
15 17 19 21 23 25 27 29
-0.5
y = 0.0612x - 0.8933
-1
R2 = 0.1027
ln(Total Capital)
Total Output
2
y = -0.002x3 + 0.1473x2 - 3.5455x + 28.346
1.5 R2 = 0.0963
1
0.5
ln(TFP)
0
15 17 19 21 23 25 27 29
-0.5
y = 0.0642x - 0.9469
-1
R2 = 0.0901
ln(Total GDP)
48
Figure 10: Relation between TFP and measures of activity in Colombian firms
Figures show the partial relations between a firm's log TFP and the level of capital per worker, total capital, and total output of the industry where the
firm's operates. Each figure also shows the results obtained in two regression of the (log) firm's TFP and each corresponding measure of aggregate
activity (also in logs). The first regression includes only the linear term while the second fits a cubic polynomial. Each regression also included a firm,
industry, and year fixed effect
8 Linear: y = -0.18x R2=0.93
Cubic: y = 1.77x + -0.65x^2 + 0.07x^3 R2=0.93
6
4
P)FT(g
lo 2
0
-2
-1 -.5 0 .5
log(Capital per capita)
log(TFP) Linear regression
Cubic polinomial
8 Linear: y = -0.23x R2=0.93
Cubic: y = 0.24x + -0.08x^2 + 0.00x^3 R2=0.93
6
) 4
PFT
og(l 2
0
-2
-.5 0 .5
log(Total Capital)
log(TFP) Linear regression
Cubic polinomial
8 Linear: y = -0.17x R2=0.93
Cubic: y = 0.58x + -0.07x^2 + 0.00x^3 R2=0.93
6
) 4
TFP(gol 2
0
-2
-1 -.5 0 .5
log(Total Output)
log(TFP) Linear regression
Cubic polinomial
49